I applied this as suggested with

stars_layer.motion_offset = Vector2(0, bg_offset)

And nope. Did not work. This still made the generic TextureRect image I applied to move, but the Shader stayed absolutely still.

And when the textureRect moved to far (i.e. reached its edge in the camera display area) the shader stayed there, but was clipped by the edge.

Sorry.. Dulviu it didn't work.

And I will continue looking for an answer.

Note: I tested this with the Shader provided above and the own shader I wanted to use similarily, and both had the same problem.